The trigonometric equation tan–1x = 3tan–1 a has solution for ______.

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The trigonometric equation tan–1x = 3tan–1a has solution for underlinebb(|a| < 1/sqrt(3)). Explanation: To solve the equation tan–1(x) = 3tan–1(a), we use the tangent function properties and transformations. Let $\theta$ = tan–1(a). Then: x = tan(3$\theta$) Using the triple-angle formula for tangent: tan(3$\theta$) = `(3tan($\theta$) - tan^3($\theta$))/(1 - 3tan^3($\theta$))` Since tan($\theta$) = a, substituting a in gives us: x = (3a - a^3)/(1 - 3a^2) For the function tan⁡−1(x) = 3tan−1(a) to have a solution, the argument of tan (which is 3$\theta$) must be within the range of the tan function, which is (-$\pi$/2, $\pi$/2). Therefore, 3$\theta$ must also be (-$\pi$/2, $\pi$/2). Given that $\theta$ = tan−1(a) is within (-$\pi$/2, $\pi$/2), the condition for 3$\theta$ to remain in this interval is: -$\pi$/6 < $\theta$ < $\pi$/6 This translates to: -$\pi$/6 < tan^-1(a) < $\pi$/6 Taking the tangent of the bounds: -1/sqrt(3) < a < 1/sqrt(3) Thus, the condition for a is: |a| < 1/sqrt(3)

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